Function approximation

University of Hamburg

MIN Faculty

Department of Informatics

Function approximationAlgorithmic Learning 64-360, Part 3d

Jianwei Zhang

University of HamburgMIN Faculty, Dept. of Informatics

Vogt-Kolln-Str. 30, D-22527 Hamburgzhang@informatik.uni-hamburg.de

27/06/2012

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MIN Faculty

Introduction Function approximation

Approximation

Approximation of the relation between x and y (curve, plane,hyperplane ) with a different function, given a limited number ofdata points D = {xi , yi}li=1.

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Approximation vs. Interpolation

A special case of approximation is interpolation:the model exactly matches all data points.

If many data points are given or measurement data is affected by noise,approximation is preferably used.

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MIN Faculty

Approximation without Overfitting

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Interpolation with Polynomials

Polynomial interpolation:

I Lagrange polynomial,

I Newton polynomial,

I Bernstein polynomial,

I Basis-Splines.

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Lagrange interpolation

To match l + 1 data points (xi , yi ) (i = 0, 1, . . . , l) with a polynomial ofdegree l , the following approach of LAGRANGE can be used:

pl(x) =l∑

yiLi (x)

The interpolation polynomial in the Lagrange form is defined as follows:

Li (x) =(x − x0)(x − x1) · · · (x − xi−1)(x − xi+1) · · · (x − xl)

(xi − x0)(xi − x1) · · · (xi − xi−1)(xi − xi+1) · · · (xi − xl)

{1 if x = xi0 if x 6= xi

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Newton Interpolation

The Newton basis polynomials of degree l are constructed as follows:

pl(x) = a0+a1(x−x0)+a2(x−x0)(x−x1)+· · ·+al(x−x0)(x−x1) · · · (x−xl−1)

This approach enables us to calculate the coefficients easily.For n = 2 the following system of equations is obtained:

p2(x0) = a0 = y0p2(x1) = a0 + a1(x1 − x0) = y1p2(x2) = a0 + a1(x2 − x0) + a2(x2 − x0)(x2 − x1) = y2

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MIN Faculty

Interpolation with Bernstein polynomials - I

Interpolation of two points with Bernstein polynomials:

y = x0B0,1(t) + x1B1,1(t) = x0(1− t) + x1t

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Interpolation with Bernstein polynomials - II

Interpolation of three points with Bernstein polynomials:

y = x0B0,2(t) + x1B1,2(t) + x2B2,2(t) = x0(1− t)2 + x12t(1− t) + x2t2

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Interpolation with Bernstein polynomials - III

Interpolation of four points with Bernstein polynomials:

y = x0B0,3(t) + x1B1,3(t) + x2B2,3(t)x3B3,3(t)

= x0(1− t)3 + x13t(1− t)2 + x23t2(1− t) + x3t3

Zhang 10

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Interpolation with Bernstein polynomials - IV

The Bernstein polynomials of degree k + 1 are defined as follows:

Bi,k(t) =

)(1− t)k−i t i , i = 0, 1, . . . , k

Interpolation with Bernstein polynomials Bi,k :

y = x0B0,k(t) + x1B1,k(t) + · · ·+ xkBk,k(t)

Zhang 11

MIN Faculty

B-Splines

A normalized B-Splines Ni,k of degree k is defined as follows: For k = 1,

Ni,k(t) =

{1 : for ti ≤ t < ti+1

0 : else

and for k > 1, the recursive definition:

Ni,k(t) =t − ti

ti+k−1 − tiNi,k−1(t)+

ti+k − t

ti+k − ti+1Ni+1,k−1(t)

with i = 0, . . . ,m.

Zhang 12

MIN Faculty

B-Spline-Curve

A B-Spline-Curve of degree k is a composite function built piecewisefrom basis B-Splines resulting in a polynomial of degree (k − 1) that is(k-2)-times continuously differentiable (class C k−2) at the borders of thesegments.The Curve is constructed by polynomials, that are defined by thefollowing parameters:

t = (t0, t1, t2, . . . , tm, tm+1, . . . , tm+k),

I m: depending on the number of data-points

I k: the fixed degree of the B-Spline curve

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Examples of B-Splines

B-Splines with degree 1, 2, 3 and 4:

tt j j+1 j+2 j+3 j+4tttt

Between the interval of parameters k B-Splines are overlapping.

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Examples of cubic B-Splines

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B-Splines of degree k - I

The recursive definition procedure of a B-Spline basis functionNi ,k(t):

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B-Splines of degree k - II

Current segments of B-Spline basis functions of degree 2, 3 and 4 for

ti ≤ t < ti+1:

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Uniform B-spline of degree 1 to 4

k=1 k=2

k=3 k=4

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Non-uniform B-Splines

Degree 3:

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Properties of B-splines

Partition of unity:∑k

i=0 Ni ,k(t) = 1.

Positivity: Ni ,k(t) ≥ 0.

Local support: Ni ,k(t) = 0 for t /∈ [ti , ti+k ].

C k−2 continuity: If the knots {ti} different in pairsthen Ni ,k(t) ∈ C k−2,i.e. Ni ,k(t) is (k − 2) times continuouslydifferentiable.

Zhang 20

MIN Faculty

Construction of B-spline curves

A B-spline curve can be constructed blending a number of predefinedvalues (data-points) with B-splines

r(t) =m∑j=0

vj ·Nj,k(t)

where vj are called control points (de Boor-points).

Let t be a given parameter, then r(t) is a point of the B-spline curve.

If t varies from tk−1 to tm+1, then r(t) is a (k-2)-times continuously

differentiable function (class C k−2).

Zhang 21

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Calculation of control points from data points

The points vj are only identical with the data points if k = 2(interpolation/otherwise approximation). The control points form aconvex hull of the interpolation curve. Two methods for thecalculation of control points from data points:

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Calculation of control points from data points

1 Solving the following system of equations (Bohm84):

qj(t) =m∑j=0

vj · Nj ,k(t)

where qj are the data-points for interpolation/approximation,j = 0, · · · ,m.

2 Learning based on gradient descent(Zhang98).

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Lattice - I

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Lattice - II

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Tensor 2D-NURBS

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Real-world Problems

I modeling: learning from examples, self-optimized formation,prediction, ...

I control: perception-action cycle, state control, Identification ofdynamic systems,...

Function approximation as a benchmark for the choice of a model

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Function approximation - 1D example

An example function f (x) = 8sin(10x2 + 5x + 1) with −1 < x < 1and the correctly distributed B-Splines:

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Lattice

The B-spline model – a two-dimensional illustration.

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Lattice (cont.)

Every n-dimensional square (n > 1) is covered by the j th

multivariate B-spline N jk(x). N j

k(x) is defined by the tensor of nunivariate B-splines:

N jk(x) =

n∏j=1

N jij ,kj

(xj) (1)

Therefore the shape of each B-spline, and thus the shape ofmultivariate ones (Figure 2), is implicitly set by their order andtheir given knot distribution on each input interval.

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Lattice (cont.)

(a) Tensor of two, order2 univariate B-splines.

(b) Tensor of one order3 and one order 2 univa-riate B-splines.

(c) Tensor of two univa-riate B-splines of order3.

Bivariate B-splines formed by taking the tensor of two univariateB-splines.

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General requirements for an approximator

I Universality: Approximation of arbitrary functions

I Generalization: good approximation without Overfitting

I Adaptivity: on the basis of new data

I Parallelism: Computing based on biological models

I Interpretability: at least “Grey-box” instead of “Black-box”

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Importance of the Interpretability of a Model

Richard P. Feynman: “the way we have to describe nature isgenerally incomprehensible to us”.

Albert Einstein: “it should be possible to explain the laws ofphysics to a barmaid”.

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Importance of the Interpretability of a Model (cont.)

Important reasons for the symbolic interpretability of anapproximator:

I Linguistic modeling is a basis of skill transfer from an expert toa computer or robot .

I Automated learning of a transparent model facilitates theanalysis, validation and monitoring in the development cycle ofa model or a controller.

I Transparen models provide diverse applications inDecision-Support Systems.

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B-Spline ANFIS

In a B-Spline ANFIS with n inputs x1, x2, . . . , xn, the rules are usedthe following form:{Rule(i1, i2, . . . , in): IF (x1 IS N1

i1,k1) AND (x2 IS N2

i2,k2) AND . . .

AND (xn IS Nnin,kn

) THEN y IS Yi1i2...in},where

I xj : input j (j = 1, . . . , n),

I kj : degree of B-spline basis function for xj ,

I N jij ,kj

: with the i-th linguistic term for the xj -associated B-splinefunction,

I ij = 0, . . . ,mj , partitioning of input j ,

I Yi1i2...in : control points for Rule(i1, i2, . . . , in).

I the “AND”-operator: product

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B-Spline ANFIS (cont.)

Then the output y of the MISO control system is:

m1∑i1=1

mn∑in=1

(Yi1,...,in

n∏j=1

N jij ,kj

This is a general B-spline model that represents the hyperplane ( itNUBS (nonuniform B-spline)).http://equipe.nce.ufrj.br/adriano/fuzzy/transparencias/anfis/anfis.pdf

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Architecture of B-Spline ANFIS

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MF(membership function)-Formulation - Tensor

Tensor of 2D-Splines:

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The activation of MF by the inputs

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B-Spline ANFIS: example

An example with two input variables (x und y) and one Output z .

The parameters of the THEN-clauses are Z1,Z2,Z3,Z4.

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B-Spline ANFIS: example (cont.)

The linguistic terms of inputs (IF-clauses):

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B-Spline ANFIS: example (cont.)

The parameters of the THEN-clauses:

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Example: control basis

The sample control basis consists of four rules:

Rule1) IF x is X1 and y is Y1 THEN z is Z1

2) IF x is X1 and y is Y2 THEN z is Z2

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Illustration of the fuzzy inference

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Illustration of the fuzzy inference (2)

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Zhang 48

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Algorithms for Supervised Learning - I

Let {(X, yd)} be a set of training data, where

I X = (x1, x2, . . . , xn) : the vector of input data,

I yd : the desired output for X.

The LSE is:

2(yr − yd)2, (2)

where yr is the current real output value during the training cycle.Goal is to find the parameters Yi1,i2,...,in , that minimize the error in(2)

2(yr − yd)2 ≡ MIN. (3)

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Algorithms for Supervised Learning - II

Each control point Yi1,...,in can be improved with the followinggradient descend algorithm:

∆Yi1,...,in = −ε ∂E

∂Yi1,...,in

= ε(yr − yd)n∏

N jij ,kj

(xj) (5)

where 0 < ε ≤ 1.

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MIN Faculty

Algorithms for Supervised Learning - III

The gradient descend algorithm ensures that the learning algorithmconverges to the global minimum of the LSE-function, because thesecond partial derivative of Y(i1, I2, hdots, in) is constant:

∂2Yi1,...,in

= (n∏

N jij ,kj

(xj))2 ≥ 0. (6)

This means that the LSE-function ( ref (error)) is convexY(i1, I2, dots, in is) and therefore has only one (global) minimum.

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Symbol Transformation of the Core Functions

Positive, convex core functions can be considered as Fuzzy sets, forexample:

µB(x) =1

1 +(x−5010

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Membership-functions

-5 0 5

(a) (b)

(e) (f)

(g) (h)

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Introduction to fuzzy sets

I fuzzy natural-language gradations of terms like “big”,“beautiful”, “strong” ...

I human thought and behavior models using the one-step logic:�� Driving: “IF-THEN”-clauses�� Car parking: With millimeter accuracy?

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Introduction to Fuzzy sets

I Use of fuzzy language instead of numerical description:�� brake 2.52 m before the curve

→ only in machine systems�� brake shortly before the curve

→ in natural language

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Definitions

Fuzzy: indistinctive, vague, unclear.

Fuzzy sets / fuzzy logic as a mechanism for

I fuzzy natural-language gradations of terms like “big”,“beautiful”, “strong” ...

I usage of fuzzy language instead of numerical description:.

I abstraction of unnecessary / too complex details.

I human thought and behavior models using the one-step logic.

Zhang 56

MIN Faculty

Characteristic function vs. Membership function

For Fuzzy-sets A we used a generalized characteristic function µA thatassigns a real number from [0, 1] to to each member x ∈ X — the“degree” of membership of x to the fuzzy set A:

µA : X → [0, 1]

µA is called membershop-function.

A = {(x , µA(x)|x ∈ X}

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Membership function

Characteristic of the continuous membership function

I Positive, convex functions (some important core functions).

I Subjective perception

I no probabilistic functions

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Membership function types - I

Triangle: trimf (x ; a, b, c) = max

(x − a

b − a,

c − x

c − b

Trapeze: trapmf (x ; a, b, c , d) = max

(x − a

b − a, 1,

d − x

d − c

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Membership function types - II

Gaussian: gaussmf (x ; c , σ) = e−12( x−c

B-Splines: bsplinemf (x , xi , xi+1, · · · , xi+k)

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Linguistic variables

A numeric variable has numerical values:

age = 25

A linguistic variable has linguistic values (terms):age : young

A linguistic value is a fuzzy set.

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Fuzzy-Partition

Fuzzy partition of the linguistic values “young”, “average” and“old”:

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Fuzzy Logic: inference mechanisms

A fuzzy rule is formulated as follows:“IF A THEN B”

with Fuzzy-sets A, B and the universes X , Y .

One of the most important inference mechanisms is the generalizedModus-Ponens (GMP):

Implication: IF x is A THEN y is BPremise: x is A′

Conclusion: y is B ′

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Fuzzy systems for function approximation

Basic idea:

I Description of the desired control behavior through naturallanguage, qualitative rules.

I Quantification of linguistic values by fuzzy sets.

I Evaluation by methods of fuzzy logic or interpolation.

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Fuzzy systems for function approximation

Fuzzy-rules:

”IF (a set of conditions is met)THEN(a set of consequences can be determined)“

In the premises (Antecedents) of the IF-part: linguistic variablesfrom the domain of process states;

In the conclusions (Consequences) of the THEN-part:linguistic variables from the system domain.

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Adaptive networks

Architecture:

Feedforward networks with different node functions

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Rule Extraction

The Fuzzy-Patches (Kosko):

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Rule Extraction

A Fuzzy-Rule-Patch:

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Additive Systeme

An additive fuzzy controller adds the “THEN”-Parts of the firedrules.

Fuzzy-Approximations-Rule:

An additive Fuzzy controller can approximate any continuousfunction f : X → Y if X is compact

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Optimal Fuzzy-Rule-Patches

Optimal fuzzy rule patches cover the extrema of a function:

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Projection of the ellipsoids on the input and output axis:

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The size of an ellipsoid depends on the training data.

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Visualization of the input-output space:

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An example for interpolation:

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Data cluster along the function:

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Approximation with Fuzzy-sets using the projections of extremes:

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Approximation of a 2D-function

z =f (x, y)

=3(1 − x)2e−x2−(y+1)2 − 10(x

5− x3 − y5)e−x2−y2 −

3e−(x+1)2−y2

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MIN Faculty

Derivatives of the function:dz

dx=− 6(1− x)e−x2−(y+1)2 − 6(1− x)2xe−x2−(y+1)2

− 10(1

5− 3x2) ∗ e−x2−y2

+ 20(1

5x − x3 − y5)xe−x2−y2

3(−2x − 2)e−(x+1)2−y2

dy=3(1− x)2(−2y − 2)e−x2−(y+1)2

+ 50y4e−x2−y2

+ 20(1

5x − x3 − y5)ye−x2−y2

3ye−(x+1)2−y2

Zhang 79

MIN Faculty

d dzdx

dx=36xe−x2−(y+1)2 − 18x2e−x2−(y+1)2 − 24x3e−x2−(y+1)2

+ 12x4e−x2−(y+1)2 + 72xe−x2−y2

− 148x3e−x2−y2

− 20y5e−x2−y2

+ 40x5e−x2−y2

+ 40x2e−x2−y2

3e−(x+1)2−y2

x2 − 8

3e−(x+1)2−y2

Zhang 80

MIN Faculty

d( dzdy )

dy=− 6(1− x)2e−x2−y(+1)2 + 3(1− x)2(−2y − 2)2e−x2−(y+1)2

+ 200y3e−x2−y2

− 200y5e−x2−y2

+ 20(1

5x − x3 − y5)e−x2−y2

− 40(1

5x − x3 − y5)y2e−x2−y2

3e−(x+1)2−y2

3y2e−(x+1)2−y2

Zhang 81

MIN Faculty

Global Overview of the Statistical Learning Theory - I

Let {(xi , yi )}li=1 be a set of data points/examples. We are searching for afunction f , which minimizes the following equation:

H[f ] =1

l∑i=1

V (yi , f (xi )) + λ‖f ‖2K

where V (·, ·) is a loss function and ‖f ‖2K is a norm in the Hilbert spaceH , which is defined by a positive kernel K , and λ is the regularizationparameter.

The problems in modeling, data regression and pattern classification are

each based on a kind of V (·, ·).

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Global Overview of the Statistical Learning Theory - II

1. V (yi , f (xi )) = (yi − f (xi ))2

(yi ∈ R1, V : square error function)⇒ Regularization Networks, RN

2. V (yi , f (xi )) = |yi − f (xi )|ε(yi ∈ R1, | · |ε: eine ε-unempfindliche Norm)

⇒ Support Vector Machines Regression, SVMR

3. V (yi , f (xi )) = |1− yi f (xi )|+(yi ∈ {−1, 1}, |x |+ = x fur x = 0, else |x |+ = 0)

⇒ Support Vector Machines Classification, SVMC

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Global Overview of the Statistical Learning Theory - II

1. V (yi , f (xi )) = (yi − f (xi ))2

(yi : a real number, V : square error function)⇒ Regularization Networks, RN

2. V (yi , f (xi )) = |yi − f (xi )|ε(yi : a real number, | · |ε: an ε-independent norm)

⇒ Support Vector Machines Regression, SVMR

3. V (yi , f (xi )) = |1− yi f (xi )|+(yi : -1 oder 1, |x |+ = x fur x = 0, sonst |x |+ = 0)

⇒ Support Vector Machines Classification, SVMC

For modeling and control tasks, the first definition is mostimportant.

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Universelal Function Approximation - I

A control-network can approximate all smooth functions with anarbitrary precision.The general solution of this problem is:

f (x) =l∑

ciK (x ; xi )

where ci are the coefficients.

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Universelal Function Approximation - II

Proposition of the approximation:

For any continuous function Y that is defined on the compactsubset Rn and the core function K , there is a functiony∗(x) =

∑li=1 ciK (x ; xi ) that fulfills for all x and any ε:

|Y (x)− y∗(x)| < ε

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Universal Function Approximation - IIUsing different kernel functions leads to different models:

K (x− xi ) = exp(−‖x− xi‖2) Gaussian RBF

K (x− xi ) = (‖x− xi‖2 + c2)−12 Inverse multiquadratic functions

K (x− xi ) = tanh(x · xi − θ) Multilayer perceptron

K (x− xi ) = (1 + x · xi )d Polynomial of degree d

K (x − xi ) = B2n+1(x − xi ) B-Splines

K (x − xi ) =sin(d + 1

2 )(x − xi )

sin (x−xi )2

Trigonometric polynomial

... ...

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Problems

1. “Curse of dimensionality”because of the exponentialdependency between the memory requirements and thedimension of input space.

2. Aliasing within the feature extraction

3. Not available target data (y).

4. Not available input factors.

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Learning from DJ-Data

Dow-Jones-Index: can the function be modeled?

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Example: Image Processing in Local Observation scenarios

A sequence of gray scale images of an object is acquired by the

movement along a fixed location:

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Example: Extraction of Eigenvectors

The first 6 Eigenvectors:

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Example: Eigenvectors and Eigenvalues

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Combination of Dimension Reduction with B-Spline Model

Eigenvectors can be partitioned by linguistic terms.

Such a combination of PCA and B-spline model can be considered as a

Neuro-Fuzzy model.

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The Neuro-Fuzzy Model

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The Training and Application Phases

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Function approximation - Algorithmic Learning 64-360, Part 3d · 2012-06-27 · University of...

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